Deterministically entangling two photonic, $N$-dimensional quantum systems using holonomy

We have devised a simple and direct procedure for deterministically entangling two photonic, quantum subsystems, each of which is defined in an $N$-dimensional Hilbert space, where $N \geq 2$. Our protocol makes use of quantum systems of four coupled photonic waveguides that instantiate $N$-dimensional, non-Abelian holonomies. The profound relationship between differential geometry—that is, holonomy—and adiabatic quantum systems—that are excited with multiple photons and operated within constant excitation number subspaces—is key to generating useful entangling interactions. Specifically, we have derived the matrix representation of the unitary holonomy, $\textbf{U}\left(N\right)$, given that $\left(N-1\right)$ photons are introduced to the holonomic device. This derivation is enabled by formally establishing a connection between the unitary evolution of the four-waveguide system, operating in a subspace of dark states in which the total excitation number is conserved, and the rotation of a high-dimensional angular momentum vector within a subspace in which the total angular momentum is conserved. Since this entangling mechanism exclusively utilizes linear optical elements and effects, it has possibly transformative implications for quantum computation with optics and photonics.